3.715 \(\int \frac{1}{(3+x) \sqrt [3]{1-x^2}} \, dx\)

Optimal. Leaf size=76 \[ \frac{1}{4} \log (x+3)-\frac{3}{8} \log \left (-\frac{1}{2} (1-x)^{2/3}-\sqrt [3]{x+1}\right )+\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(1-x)^{2/3}}{\sqrt{3} \sqrt [3]{x+1}}\right ) \]

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] - (1 - x)^(2/3)/(Sqrt[3]*(1 + x)^(1/3))])/4 + Log[3 + x]/4 - (3*Log[-(1 - x)^(2/3)/2
- (1 + x)^(1/3)])/8

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Rubi [A]  time = 0.0144284, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {753, 123} \[ \frac{1}{4} \log (x+3)-\frac{3}{8} \log \left (-\frac{1}{2} (1-x)^{2/3}-\sqrt [3]{x+1}\right )+\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(1-x)^{2/3}}{\sqrt{3} \sqrt [3]{x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((3 + x)*(1 - x^2)^(1/3)),x]

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] - (1 - x)^(2/3)/(Sqrt[3]*(1 + x)^(1/3))])/4 + Log[3 + x]/4 - (3*Log[-(1 - x)^(2/3)/2
- (1 + x)^(1/3)])/8

Rule 753

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> Dist[a^(1/3), Int[1/((d + e*x)*(1 - (3*e
*x)/d)^(1/3)*(1 + (3*e*x)/d)^(1/3)), x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + 9*a*e^2, 0] && GtQ[a, 0]

Rule 123

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[
(b*(b*e - a*f))/(b*c - a*d)^2, 3]}, -Simp[Log[a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[(Sqrt[3]*ArcTan[1/Sqrt[3
] + (2*q*(c + d*x)^(2/3))/(Sqrt[3]*(e + f*x)^(1/3))])/(2*q*(b*c - a*d)), x] + Simp[(3*Log[q*(c + d*x)^(2/3) -
(e + f*x)^(1/3)])/(4*q*(b*c - a*d)), x])] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]

Rubi steps

\begin{align*} \int \frac{1}{(3+x) \sqrt [3]{1-x^2}} \, dx &=\int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{1+x} (3+x)} \, dx\\ &=\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(1-x)^{2/3}}{\sqrt{3} \sqrt [3]{1+x}}\right )+\frac{1}{4} \log (3+x)-\frac{3}{8} \log \left (-\frac{1}{2} (1-x)^{2/3}-\sqrt [3]{1+x}\right )\\ \end{align*}

Mathematica [C]  time = 0.039634, size = 68, normalized size = 0.89 \[ -\frac{3 \sqrt [3]{\frac{x-1}{x+3}} \sqrt [3]{\frac{x+1}{x+3}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{4}{x+3},\frac{2}{x+3}\right )}{2 \sqrt [3]{1-x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((3 + x)*(1 - x^2)^(1/3)),x]

[Out]

(-3*((-1 + x)/(3 + x))^(1/3)*((1 + x)/(3 + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, 4/(3 + x), 2/(3 + x)])/(2*(1
- x^2)^(1/3))

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Maple [F]  time = 0.392, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{3+x}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3+x)/(-x^2+1)^(1/3),x)

[Out]

int(1/(3+x)/(-x^2+1)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x^{2} + 1\right )}^{\frac{1}{3}}{\left (x + 3\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+x)/(-x^2+1)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((-x^2 + 1)^(1/3)*(x + 3)), x)

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Fricas [B]  time = 5.34306, size = 350, normalized size = 4.61 \begin{align*} \frac{1}{4} \, \sqrt{3} \arctan \left (-\frac{18031 \, \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )} - \sqrt{3}{\left (5054 \, x^{2} + 8497 \, x + 23659\right )} - 57889 \, \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{6859 \, x^{2} - 240699 \, x - 220122}\right ) - \frac{1}{8} \, \log \left (\frac{x^{2} - 6 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )} + 6 \, x + 12 \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 9}{x^{2} + 6 \, x + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+x)/(-x^2+1)^(1/3),x, algorithm="fricas")

[Out]

1/4*sqrt(3)*arctan(-(18031*sqrt(3)*(-x^2 + 1)^(1/3)*(x - 1) - sqrt(3)*(5054*x^2 + 8497*x + 23659) - 57889*sqrt
(3)*(-x^2 + 1)^(2/3))/(6859*x^2 - 240699*x - 220122)) - 1/8*log((x^2 - 6*(-x^2 + 1)^(1/3)*(x - 1) + 6*x + 12*(
-x^2 + 1)^(2/3) + 9)/(x^2 + 6*x + 9))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x + 3\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+x)/(-x**2+1)**(1/3),x)

[Out]

Integral(1/((-(x - 1)*(x + 1))**(1/3)*(x + 3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x^{2} + 1\right )}^{\frac{1}{3}}{\left (x + 3\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3+x)/(-x^2+1)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((-x^2 + 1)^(1/3)*(x + 3)), x)